Problem: What is the period of $y=-2\sin\left(\dfrac23x-4\right)$ ? Give an exact value. units
Period in sinusoids of the form $y=a\sin(bx+c)+d$ Graphically, the period of a sinusoidal function is the horizontal distance between the ends of a single cycle of its graph. The period of a sinusoid of the form $y={a}\sin( {b}x + c) + {d}$ is equal to $\dfrac{2\pi}{| b|}$. [How can we justify this given our graphical understanding of period?] Finding the period The period of $y=-2\sin\left({\dfrac23}x-4\right)$ is: $\begin{aligned} \text{period}&=\dfrac{2\pi}{|{b}|}\\\\ &=\dfrac{2\pi}{\left|{\dfrac23}\right|} \\\\\\\\\\ &= 2\pi\cdot \dfrac{3}{2} \\\\ &=3\pi \end{aligned}$ The answer The period of $y=-2\sin\left(\dfrac23x-4\right)$ is $3\pi$ units.